Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500763 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,10)(5,8)(6,12)(7,11)(9,14)(15,29)(16,21)(17,26)(18,22)(19,25)(20,24)(23,30)(27,31)(28,32);(2,8)(3,14)(4,17)(5,6)(10,25)(11,12)(13,21)(16,20)(18,19)(23,29)(27,32)(28,30);(1,3)(2,9)(4,15)(5,14)(6,20)(7,16)(8,21)(10,23)(11,24)(12,13)(17,30)(18,28)(19,31)(22,29)(25,32)(26,27);(1,4)(2,10)(3,15)(5,18)(6,17)(7,19)(8,22)(9,23)(11,25)(12,26)(13,27)(14,28)(16,31)(20,30)(21,29)(24,32);(2,11)(8,12)(9,24)(10,25)(13,21)(22,26)(23,32)(27,29)Polycyclic presentation:
2^9|f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72216013, #72216095, … |
| maximal normal subgroup of | #72213514, #72216013, #72216095, … |
| maximal quotient of | #72250584, C2×#7500763, #72255658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407959);